GRE Triangles

Time to learn about triangles okay you have to know these rules you have to know these formulas please memorize it no in the back of your heart and know how to use this to your advantage.

Let`s go ahead and look at a triangle okay before I go ahead and give you the formula let`s go ahead and take a look at a problem.

Let`s just say I have this triangle over here I have a 45 45 90 degree triangle so here`s my right angle here is 45 degrees here is 45 degrees and I say hey the hypotenuse is 4 radical to find me the area right finding the area what is the area.

Let`s just find the area for it let`s focus on that how do you go ahead and approach this problem right I`m only given the hypotenuse 4 radical 2 I don`t know what my legs are you know seemed like a difficult question except it`s not really a difficult question at all you have to know what the formula you have to know that this is a special kind of triangle it`s a 45-45-90 triangle and basically this is this is what you have to understand about a 45-45-90 triangle it`s that the legs are represented by the variable X and the hypotenuse is represented by X radical 3.

There`s a standard you have to know that`s okay so this leg is equal to this leg and the hypotenuse is x times radical 3 so that`s how can we use this information to help us with this question over here right so let`s just say this is question number 1
Well look this up not X radical 3 X radical 2 okay so right away I`ve already made a mistake okay so pay very careful attention not extra equal 3x radical to memorize it don`t mess up like me okay X radical 2 take a look at this problem.

4 radical 2 right what it`s very similar here that`s good that means that this problem is going to be very easy I know if the hypotenuse 4 rather than 2 my legs should be X what`s X right here 4 so I know that this leg is 4 and I know that this leg is for and I want to find out the area.

The air what`s the area triangle 1/2 base times height 1/2 base times height well my base is over here which is 4 and my height is 4 actually the hypothesis not needing this problem so it will be 1/2 times 4 times 4 and you get 4 times 4 16 16 divided by 2 is 8
Now what if I asked you for the perimeter okay very simple okay you find the perimeter you add up all sides 4 plus 4 plus 4 radical 2 what do you get so P is equal to 4 plus 4 is 8 plus 4 radical 2 and you have to leave that because you can`t really mess with the radical 2 in this case they don`t combine so you want to leave it as it is ok so here is a question and this is a simple type of question.

Now what happens if I you know kind of play around with these numbers ok what happens if I gave you a problem like this ok so let`s go ahead and separate this so you don`t get confused and lost let`s say question number 2 I have another 45 45 90 degree triangle and in this case instead of 4 radical 2 I just have 4 so the value of my hypotenuse, in this case, is 4 well now it gets a little bit tricky right if you don`t know algebra that well .

Or how to manipulate what you already have here right so I know that this is X radical 2 yet I have 4 over here so I can you know it`s kind of hard from you I know that this is X and this is X and I know that this is equal to X radical 2 so what can I do actually well if it`s actually fairly simple and or if you have this and you want to find out this what you can do is divide this number over here by radical 2 right because if you go ahead and divide X radical 2 by radical 2 they cancel out and you get X and you know what we`re good here right we`re at a good set point so let`s go ahead and do that we have 4 over here so I`ll go ahead and divide 4 divided by radical 2 and now I`m left with 4 divided by radical 2 but hold on radical 2 is a disgusting number right you want to go ahead and rationalize the denominator and you want to go ahead and not leave a radical in the denominator because it`s going to make this entire thing very messy because what if I asked you to find the area right it`s going to be very messy so what do you do well go ahead and multiply both the numerator and denominator by radical two so that`s called when we say to rationalize the denominator and you basically take the conjugate, in this case, it`s just one term so it`s very simple but if I go ahead and multiply both sides by radical 2 well what happens to the denominator radical two times radical two is a radical four and radical 4 is a perfect square which turns out to be two so it`s very helpful in this situation and then just the numerator is four times radical two so I have four radical two and I can simplify this we already mentioned two – so my question my answer here is four radical two over two or my answer but this is what.

X is equal to right so X is equal to four radical two over two and of course this is also equal to four radical two over two and now if I want to find out the area right if I want to find out the area what is the area of a triangle once again area of a triangle is one half base times height the base, in this case, is this ugly number over here.
1/2 times 4 radical 2 over 2 times the height the height is this once again also one other ugly number 4 radical 2 over 2 and now you can go ahead and solve this problem radical I`m sorry 4 times 4 is 16 and then radical two times radical 2 is a radical 4 which simplifies to just two so 16 times 2 all over 2 times 2 is 4 4 times 2 is 8 and then there you go you can go ahead and simplify the problem 16 times 2 which is 32 over 8 or just 4.

That`s how you go and do this problem okay so now let`s go and take a look at another question okay what if I gave you the same triangle but now I said you know what I`m just going to go ahead and give you the perimeter the perimeter is equal to a plus 4 radical 2 8 plus 4 radical 2 and I have a 90 degree here 45 45 and I`m telling you hey I find the value of BC right by the length of BC so I want to find out what this is I want to find out what this is and I know this is a 45 45 90 degree triangle so I know that this X and this X and this becomes X or radical 2 right and now how do I go and solve this problem I want to find out the value of this I know that the perimeter is equal to 8 plus 4 radical 2 this requires some out-of-the-box thinking right well not out-of-the-box thinking but you know you really have to just understand and use the values and things that information that you`re given right and you well you should know this formula you should know that the perimeter is essentially side plus side plus side right so I mean over here I also have these values right these are not values of variables but what happens if we set these equations to each other right because they should indeed equal each other because here is a perimeter I mean given the perimeter this is the perimeter of the triangle right so I can go ahead and make it to an equation X plus X plus X radical 2 and this entire thing is equal to the perimeter and I know that the perimeter is equal to a plus 4 radical 2 so this makes perfect sense now if I go ahead and try to solve for X I can go ahead and actually find out the answer ok so I can go I have two like terms I have well I have one like term over here X plus X which is 2x plus X 4 radical 2 is equal to 8 plus 4 radical 2 and right away you can see a beautiful relationship you can see what I don`t know hopefully you`ve caught this but here is a radical 2 here`s a radical 2 well this is looking very similar over here and here is 4 so x over here is equal to 4 right well the hypotenuse in this figure is equal to 4 radical 2 and then over here well I know that here is 2 X is equal to 8 therefore I can divide 2 by both sides and X is equal to 4 so X is equal to 4 so the answer right my original question what is the length of BC that would be 4 but once again it makes total sense because Hey look at this I can redraw the triangle here is a 45 45 90 degree triangle 4 4 4 radical 2 it follows the the standard rule that we have right that in a 45 45 90 degree triangle the legs are equal to X and the hypotenuse is equal to X radical 2 so this is how you approach this kind of problem so now let`s go ahead and take a look at a 30 60 90 degree triangle we`ve already looked at several examples of you here for a 45 45 90 degree triangle so hopefully your action you`re actually very well familiar with these kinds of questions already so let`s go ahead and take a look at a 30 60 90 degree triangle I do apologize for my triangles not being straight so here`s 30 degrees and here is 60 degrees and let`s go and take up the initial information that you have to know you have to know that this side right the side that is opposite of 30 degrees is equivalent to X right so this side is equal to X this side is equal to X radical 3 right this the these six degrees opposite side is X radical 3 and then the hypotenuse would be 2x.

It`s brittle bit different you should memorize this you should know this you should know how to use this to your advantage of course when given a question ok so let`s go ahead and I won`t give you such a simple problem now well it`s still simple right let`s so let`s just say I have another 30 60 90 degree triangle whoops 30s and here`s 60 30 60 90 degree triangle and I say the shortest side the shortest side is equal to 5 right the length of the shortest side is 5 well how do you go ahead and do that well you should already know that this is the shortest side if you don`t know that well here is the smallest angle 30 degrees so the length opposite of 30 degrees will represent the shortest side so in this case this side will be 5.

Let`s just say I want to find out the area okay I can find out the area but on the perimeter I`m just just just making up a question of here what is the area of this so here`s a 30-degree triangle the opposite side the shortest side is 5 and now I`m kind of stuck right if I want to find out the area well I know that the area is equal to one half base times height so at minimum I need to find out this value which I don`t know so this is not a 45 45 90 degree triangle where it`s so simple because in you know this five will be a five here instead over here you have to follow this rule so instead of just five it`s now 5 radical 3 so it`s 5 radical 3 and now automatically I know let`s just go ahead and do the hypotenuse for fun.

The hypotenuse is 2x that means X is 5 5 times 2 is 10 so the high part of the hypotenuse is 10 and now if I want to solve for my original question a is equal to 1/2 base times height 1/2 times the base is 5 radical 3 so times 5 radical 3 times the height which is 5 ok so I have that 1/2 times 5 radical 3 times 5 and I can go in and combine that so 5 times 5 is 25 25 radical 3 all over 2 so that is my answer the which is the area of this figure over here.

If you want to find the perimeter go ahead and try that yourself for the perimeter to go ahead and try well not try but add up all these lengths over here so 5 plus 5 radical 3 plus 10 and then you can find the perimeter let`s go ahead take a look at another question let`s let`s go ahead and take a look at an important rule that you have to know let`s draw it over here I guess hopefully you can see it hopefully I have enough space let`s just say I have this triangle and no it`s not equilateral although it looks like a lateral forget about it the skills not on this this this diagram is not drawn to scale and let`s just say that this value is 11 this value is let`s say 7 ok and now I`m going to ask you
To find out the perimeter of this triangle so I`m asking you to find the perimeter of this triangle given that this side is an integer and it is the greatest possible integer so forgive me that I didn`t write this question down I`m just making it up as I go along ok so I`ll repeat that if you want to write that down.

Here`s a question I want to find the perimeter given that this length is going to be an integer but the greatest possible integer okay and why am I doing this why am i why am i asking that because you have to know this kind of question or this question appears frequently and is tested basically so this is x over here in saying you have to go ahead and know this rule that this side cannot be less than the difference of these two sides or greater than the sum of those both sides so what does that mean well 11 minus 7 is 3 right look for 11 minus 7 is 4 that means that X cannot be less than 4 so X cannot be well X is greater than 4.

X is greater than 4 but 11 plus 7 which is 18 meaning that X is less than 18 okay that`s the rule so my question states that I have to have the largest possible integer and the largest possible integer well it cannot be 18 right because it`s not X is less than or equal to it`s just X is less than 18 so the greatest possible integer this can be is 17 so over here I know that X is equal to 17 and now I can find my perimeter 11 plus 7 plus 17 I can go ahead and solve for that and that would be your answer ok so make sure you know this type of problem or more specially know the rule that when you`re given two sides that`s the third side of the triangle cannot be less than the difference of the two other of the two other sides or greater than the sum of the two other sides it`s very important

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